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Theorem conncompss 21645
Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompss ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1127 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑋)
2 conntop 21629 . . . . . . 7 ((𝐽t 𝑇) ∈ Conn → (𝐽t 𝑇) ∈ Top)
323ad2ant3 1126 . . . . . 6 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐽t 𝑇) ∈ Top)
4 restrcl 21369 . . . . . . 7 ((𝐽t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
54simprd 491 . . . . . 6 ((𝐽t 𝑇) ∈ Top → 𝑇 ∈ V)
6 elpwg 4386 . . . . . 6 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
73, 5, 63syl 18 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
81, 7mpbird 249 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9 3simpc 1143 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn))
10 eleq2 2847 . . . . . 6 (𝑦 = 𝑇 → (𝐴𝑦𝐴𝑇))
11 oveq2 6930 . . . . . . 7 (𝑦 = 𝑇 → (𝐽t 𝑦) = (𝐽t 𝑇))
1211eleq1d 2843 . . . . . 6 (𝑦 = 𝑇 → ((𝐽t 𝑦) ∈ Conn ↔ (𝐽t 𝑇) ∈ Conn))
1310, 12anbi12d 624 . . . . 5 (𝑦 = 𝑇 → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) ↔ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
14 eleq2 2847 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
15 oveq2 6930 . . . . . . . 8 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
1615eleq1d 2843 . . . . . . 7 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
1714, 16anbi12d 624 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1817cbvrabv 3395 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)}
1913, 18elrab2 3575 . . . 4 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
208, 9, 19sylanbrc 578 . . 3 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
21 elssuni 4702 . . 3 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
2220, 21syl 17 . 2 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
23 conncomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2422, 23syl6sseqr 3870 1 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  {crab 3093  Vcvv 3397  wss 3791  𝒫 cpw 4378   cuni 4671  (class class class)co 6922  t crest 16467  Topctop 21105  Conncconn 21623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-rest 16469  df-top 21106  df-conn 21624
This theorem is referenced by:  conncompcld  21646  tgpconncompeqg  22323  tgpconncomp  22324
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